Ruelle operator for infinite conformal iterated function systems

Let X,{wj}j=1m,{pj}j=1m(2⩽m<∞) be a contractive iterated function system (IFS), where X   is a compact subset of RdRd. It is well known that there exists a unique nonempty compact set K   such that K=⋃j=1mwj(K). Moreover, the Ruelle operator on C(K  ) determined by the IFS X,{wj}j=1m,{pj}j=1m(2⩽m<∞) has been extensively studied. In the present paper, the Ruelle operators determined by the infinite conformal IFSs are discussed. Some separation properties for the infinite conformal IFSs are investigated by using the Ruelle operator.